Retrieval of source parameters of an event of the 2000 West Bohemia earthquake swarm assuming an anisotropic crust

We propose an inversion scheme for retrieval of characteristics of seismic point sources, which in contrast to common practice, takes into account anisotropy. If anisotropy is neglected during inversion, the moment tensors retrieved from seismic waves generated by sources situated in anisotropic media may be biased. Instead of the moment tensor, the geometry of the source is retrieved directly in our inversion; if necessary, the moment tensor can be then determined from the source geometry aposteriori. The source geometry is defined by the orientation of the slip vector and the fault normal as well as the strength of the event given by the size of the slip and the area of the fault. This approach allows direct interpretation of the source geometry in terms of shear and tensile faulting. It also makes possible to identify volumetric source changes that occur during rupturing. We apply the described algorithm to one event of the 2000 West Bohemia earthquake swarm episode. For inversion we use information of the direct P waves. The structure is approximated by three different models determined from travel-time observations. The models are inhomogeneous isotropic, inhomogeneous anisotropic, and homogeneous anisotropic. For these models we obtain seismic moments M_T = 3.2 − 3.8 × 10¹⁴ Nm and left-lateral near-vertical oblique normal faulting on a N-S trending rupture surface. The orientation of the rupture surface is consistent with fault-plane solutions of earlier studies and with the spatial distribution of other events during this swarm. The studied event seems to be accompanied by a small amount of crack opening. The amount of crack opening is slightly reduced when the inhomogeneous anisotropic model is assumed, but it persists. These results and additional independent observations seem to indicate that tensile faulting occurs as a result of high fluid pressure.     

Microseismic Full Waveform Modeling in Anisotropic Media with Moment Tensor Implementation

Seismic anisotropy which is common in shale and fractured rocks will cause travel-time and amplitude discrepancy in different propagation directions. For microseismic monitoring which is often implemented in shale or fractured rocks, seismic anisotropy needs to be carefully accounted for in source location and mechanism determination. We have developed an efficient finite-difference full waveform modeling tool with an arbitrary moment tensor source. The modeling tool is suitable for simulating wave propagation in anisotropic media for microseismic monitoring. As both dislocation and non-double-couple source are often observed in microseismic monitoring, an arbitrary moment tensor source is implemented in our forward modeling tool. The increments of shear stress are equally distributed on the staggered grid to implement an accurate and symmetric moment tensor source. Our modeling tool provides an efficient way to obtain the Green’s function in anisotropic media, which is the key of anisotropic moment tensor inversion and source mechanism characterization in microseismic monitoring. In our research, wavefields in anisotropic media have been carefully simulated and analyzed in both surface array and downhole array. The variation characteristics of travel-time and amplitude of direct P- and S-wave in vertical transverse isotropic media and horizontal transverse isotropic media are distinct, thus providing a feasible way to distinguish and identify the anisotropic type of the subsurface. Analyzing the travel-times and amplitudes of the microseismic data is a feasible way to estimate the orientation and density of the induced cracks in hydraulic fracturing. Our anisotropic modeling tool can be used to generate and analyze microseismic full wavefield with full moment tensor source in anisotropic media, which can help promote the anisotropic interpretation and inversion of field data.     

On the retrieval of moment tensors from borehole data

The complete moment tensors of seismic sources in homogeneous or vertically inhomogeneous isotropic structures cannot be retrieved using receivers deployed in one vertical borehole. The complete moment tensors can be retrieved from amplitudes of P‐waves, provided that receivers are deployed in at least three boreholes. Using amplitudes of P‐ and S‐waves, two boreholes are, in principle, sufficient. Similar rules also apply to transversely isotropic media with a vertical axis of symmetry. In the case of limited observations, the inversion can be stabilized by imposing the zero‐trace constraint on the moment tensors. However, this constraint is valid only if applied to observations of shear faulting on planar faults in isotropic media, which produces double‐couple mechanisms. For shear faulting on non‐planar faults, for tensile faulting, and for shear faulting in anisotropic media, the zero‐trace constraint is no longer valid and can distort the retrieved moment tensor and bias the fault‐plane solution. Numerical modelling simulating the inversion of the double‐couple mechanism from real data reveals that the errors in the double‐couple and non‐double‐couple percentages of the moment tensors rapidly decrease with increase in the number of boreholes used. For noisy P‐ and S‐wave amplitudes with noise of 15% of the top amplitude at each channel and for a velocity model biased by 10%, the errors in the double‐couple percentage attain 25, 13 and 6% when inverting for the double‐couple mechanism from one, two and three boreholes.     

Moment tensor inversion of near-source broadband data

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The role of seismic methods in investigation of rock mass

The existence of cracks in a rock mass causes the reduction of seismic wave velocity and the anisotropy of seismic wave velocity, which is characteristic for rocks with preferred orientation of cracks. The present study concerns sedimentary rocks (sandstone, limestone) and igneous rocks (basalt, granite). Studying the relationship between seismic anisotropy and cracks anisotropy in rocks I was only interested in fractures perpendicular to the layering. This allowed me to calculate two-dimensional crack tensors and velocity tensors in planes parallel to the layer surface. An application of tensor calculus enables to take into account both geometry and orientation of cracks. The obtained results confirm that the directions of major axes of second-rank velocity tensor prove the relationship consistent with predictions of theoretical models. Owing to these dependences, seismic methods can be used to study the cracks anisotropy in rocks inaccessible for direct observations.     

非对称地震矩张量时间域反演:理论与方法

本文在对称地震矩张量反演的基础上,进一步研究了非对称地震矩张量时间域反演的理论与方法,结果表明:非对称地震矩张量反演与对称地震矩张量反演类似,只需将对称地震矩张量反演方法略加改动,即增加3个待解参数,便可实现非对称地震矩张量反演.为了判断非对称地震矩张量反演相对于对称地震矩张量反演是否存在过度拟合,运用了AIC准则 (赤池信息准则).为了定量地描述地震矩张量之间的差异,引入了地震矩张量的矢量表示法.通过分析格林函数与地震矩张量各分量之间的关系,得出:在非对称地震矩张量反演时,若仅用垂直向地动位移数据,将无法区分M_xy与M_yx这两个分量, 需要同时运用垂直向与水平向地动位移数据进行联合反演才能区分M_xy与M_yx; 若采用不同的速度结构模型或不同的格林函数计算方法,则需重新评估地震矩张量各分量的分辨度问题.为检验非对称地震矩张量反演方法的可行性, 利用合成地震图进行了一系列数值试验.数值试验结果表明,在非对称地震矩张量反演中,有必要引入S波进行P波与S波联合反演以提高反演的准确性和判定断层面的能力.      

Point source radiation in inhomogeneous anisotropic structures

The ray formulae for the radiation from point sources in unbounded inhomogeneous isotropic as well as anisotropic media consist of two factors. The first one depends fully on the type and orientation of the source and on the parameters of the medium at the source. We call this factor the directivity function. The second factor depends on the parameters of the medium surrounding the source and this factor is the well-known geometrical spreading. The displacement vector and the radiation pattern defined as a modulus of the amplitude of the displacement vector measured on a unit sphere around the source are both proportional to the ratio of the directivity function and the geometrical spreading.For several reasons it is desirable to separate the two mentioned factors. For example, there are methods in exploration seismics, which separate the effects of the geometrical spreading from the observed wave field (so-called true amplitude concept) and thus require the proposed separation. The separation also has an important impact on computer time savings in modeling seismic wave fields generated by point sources by the ray method. For a given position in a given model, it is sufficient to calculate the geometrical spreading only once. A multitude of various types of point sources with a different orientation can then be calculated at negligible additional cost.In numerical examples we show the effects of anisotropy on the geometrical spreading, the directivity and the radiation pattern. Ray synthetic seismograms due to a point source positioned in an anisotropic medium are also presented and compared with seismograms for an isotropic medium.     

The impact of different clay minerals on the anisotropy of clay matrix in shale

Although clay is composed of disconnected anisotropic clay platelets, many rock physics models treat the clay platelets in shale as interconnected. However, the clay matrix in shales can be modelled as anisotropic clay platelets embedded within a soft isotropic interplatelet region, allowing the influence of disconnected clay platelets on the elastic properties of the clay matrix to be analysed. In this model, properties of the interplatelet region are governed by its effective bulk and shear moduli, whereas the effective properties of the clay platelets are governed by their volume fraction, aspect ratio and elastic stiffness tensor. Together, these parameters implicitly account for variations in clay and fluid properties, as well as fluid saturation. Elastic stiffnesses of clay platelets are obtained from the literature, including both experimental measurements and first-principles calculations of the full anisotropic (monoclinic or triclinic) elastic stiffness tensors of layered silicates. These published elastic stiffness tensors are used to compile a database of equivalent transverse isotropic elastic stiffness tensors, and other physical properties, for eight common varieties of layered silicates. Clay matrix anisotropy is then investigated by examining the influence of these different elastic stiffnesses, and of varying model parameters, upon the effective transverse isotropic elastic stiffness tensor of the clay matrix. The relationship between the different clay minerals and their associated anisotropy parameters is studied, and their impact on the resulting anisotropy of the clay matrix is analysed.     

Effects of 1-D versus 3-D velocity models on moment tensor inversion in the Dobrá Voda area in the Little Carpathians region,Slovakia

Retrieving the parameters of a seismic source from seismograms involves deconvolving the response of the medium from seismic records. Thus, in general, source parameters are determined from both seismograms and the Green functions describing the properties of the medium in which the earthquake focus is buried. The quality of each of these two datasets is equally significant for the successful determination of source characteristics. As a rule, both sets are subject to contamination by effects that decrease the resolution of the source parameters. Seismic records are generally contaminated by noise that appears as a spurious signal unrelated to the source. Since an improper model of the medium is quite often employed, due to poor knowledge of the seismic velocity of the area under study, and since the hypocentre may be mislocated, the Green functions are not without fault. Thus, structures not modelled by Green functions are assigned to the source, distorting the source mechanism. To demonstrate these effects, we performed a synthetic case study by simulating seismic observations in the Dobrá Voda area of the Little Carpathians region of Slovakia. Simplified 1-D and 3-D laterally inhomogeneous structural models were constructed, and synthetic data were calculated using the 3-D model. Both models were employed during a moment tensor inversion. The synthetic data were contaminated by random noise up to 10 and 20 % of the maximum signal amplitude. We compared the influence of these two effects on retrieving moment tensors, and determined that a poor structural model can be compensated for by high-quality data; and that, in a similar manner, a lack of data can be compensated for by a detailed model of the medium. For examples, five local events from the Dobrá Voda area were processed.     

Decomposition of seismic moment tensors for underground nuclear explosions

Generally the decomposition of a seismic moment tensor is not unique. However, to favorably view the characteristics of a certain seismic source, one must decompose a seismic moment tensor into parts according to assumptions about the properties of the seismic source. Different from natural earthquakes in which the shear dislocation component plays a predominant role in the source process, and the seismic moment tensor can be separated into an isotropic component, a double couple, and a compensated linear vector dipole (CLVD), underground nuclear explosions have three major components in their source process, i.e., the explosion, the tensional spalling, and the tectonic strain release associated with the explosion. In such a situation the conventional moment tensor decomposition for earthquakes is not convenient to estimate the yield of the explosion and to characterize the tectonic strain release. In this paper, an alternative decomposition scheme is proposed to deal with the moment tensor of underground nuclear explosions, which might benefit the approach to study the tectonic strain release induced by underground nuclear detonations.     

Shear-tensile crack as a tool for reliable estimates of the non-double-couple mechanism: West Bohemia-Vogtland earthquake 1997 swarm

Shear-tensile crack is a model for an earthquake mechanism that is more constrained than the moment tensor but that can still describe a non-shear focus. As such, the shear-tensile crack model is more robust than the moment tensor model and yields more reliable estimates for the earthquake mechanism. Such an advantage verifies the credibility of the non-double-couple component found for some events of the 1997 West Bohemia-Vogtland earthquake swarm. As expected, in several cases, a significantly resolved non-double-couple component was obtained where the moment tensor approach failed. Additionally, for non-shear sources, the shear-tensile crack model offers optimization of the Poisson number within the focus, concurrently with retrieval of the mechanism. However, results obtained for the joint inversion of the 1997 swarm indicate that resolution is low. A series of synthetic experiments indicated that limited observations during 1997 were not the cause. Rather, hypothetical experiments of both very good and extremely poor network configurations similarly yielded a low resolution for the Poisson number. Applying this method to data for recent swarms is irrelevant because the small non-double-couple components detected within the inversion are spurious and, thus, the events are pure double-couple phenomena.     

Source Mechanisms of Induced Earthquakes at The Geysers Geothermal Reservoir

At The Geysers geothermal reservoir in northern California, evidence strongly suggests that activities associated with production of electric power cause an increase in the number of small earthquakes. First-degree dynamic moment tensors are used to investigate the relationship between induced earthquakes and injection of water into a well as part of a controlled experiment in the northwest Geysers. The estimation of dynamic moment tensors in the complex shallow crust at The Geysers is challenging, so the method is described in detail with particular attention given to the uncertainty in the results. For seismic events in the moment magnitude range of 0.9–2.8, spectral moduli of dynamic moment tensors are reliably recovered in the frequency range of 1–100 Hz, but uncertainty in the associated spectral phases limits their use to a few simple results. A number of different static moment tensors are investigated, with the preferred one obtained from parameters of a model fitted to the spectral modulus of the dynamic moment tensor. Moment tensors estimated for a group of 20 earthquakes exhibit a range of source mechanisms, with over half having significant isotropic parts of either positive or negative sign. Corner frequencies of the isotropic part of the moment tensor are about 40 % larger than the average of the deviatoric moment tensor. Some spatial patterns are present in source mechanisms, with earthquakes closely related in space tending to have similar mechanisms, but at the same time, some nearby earthquakes have very different mechanisms. Tensional axes of displacement in the source regions are primarily horizontal, while the pressure axes range from near horizontal to vertical. Injection of water into the well in the center of the study area clearly causes an increase in the number of earthquakes per day, but an effect upon source mechanisms is not evident.     

三峡库区秭归与巴东交界Ms4.0地震矩张量及应力环境研究

三峡库区湖北秭归与巴东交界地区2013年以来发生5次M S4.0以上地震.本文基于湖北、重庆等省属13个测震台站宽频带波形数据,采用kiwi软件包和rapidinv12程序反演了这些地震的全矩张量解,结果显示所有地震双力偶解均为走滑兼逆断性质,两个节面走向、倾角、滑动角与P、T轴走向和倾俯角的一致性均较好,主压应力P轴NWW走向,倾角近水平,主张应力T轴NNE走向.与三峡库区现今构造应力场比较,这5次地震主压应力P轴与主张应力T轴的平均方向与其大体一致,并且节面1平均走向也与附近主要断裂:高桥断裂和周家山—牛口断裂走向大体一致,反映了这些地震受到区域现今构造应力环境的控制与影响.通过对地震矩张量的ISO+DC+CLVD分解显示,所有地震的ISO成分均低于10%,纯双力偶DC成分偏少而CLVD成分较多,主要与震源区地下浅层介质复杂的各向异性、多重剪切错动或液态下拉张错动有关.结合震源区局部特殊的岩层岩性结构和水文地质环境,分析认为库水长时期的侵蚀溶蚀与加卸荷载作用,有利于引起局部岩层强度降低,在特殊水位时段将触发岩层失稳而滑动破裂,这5次地震中CLVD成分较多正是这一复杂过程的重要体现.     

Moment Tensor Solutions and Triggering Environment for Earthquakes in Koyna-Warna Water Reservoirs Region,India

We consider nine earthquakes in the Koyna-Warna reservoir region on the western side of the Peninsular India. The deviatoric moment tensors of these earthquakes have been evaluated by minimizing the least-squares misfit between observed and synthetic seismograms. We use broadband seismograms of observatories at KARD and PUNE which are at distances of nearly 50 and 150 km, respectively, from the epicenters. Both surface wave inversion and the difference between the arrival times of SH and SV show the presence of an anisotropic crust. However, we have obtained an equivalent isotropic structure by improving the published crustal structures of this area through inversion of surface wave group velocity data. The deviatoric moment tensors of the earthquakes are decomposed into two components: double-couple and compensated linear vector dipoles (CLVD). The double-couple components of all the nine earthquakes show normal faulting with minor strike slip; the T axis is consistently subhorizontal with an average azimuth of 260.6° and the P axis is nearly vertical. The fault planes of six events give average strike direction and dip, respectively as 194.0° and 51.8° and are associated with the main fault of the area. The other three events lie in the southern part of this area and have strike direction between SSE and SE which is parallel to the tectonic features in this part. The CLVD component is generally within 20 percent of the total moment tensor. Recent studies show that anisotropy can produce source mechanism with CLVD up to 30 percent and can also cause high pore fluid pressure leading to fault instability more rapidly compared to conventional mechanism in an isotropic medium. It appears that the anisotropic crust, noted in the present work, is generating the CLVD component and also gives the proper environment to trigger earthquakes by reservoirs through pore fluid pressure.     

Inferring the Orientation-distribution Function from Observed Seismic Anisotropy: General Considerations and an Inversion of Surface-wave Dispersion Curves

—A general relation linking the elasticity tensor of an anisotropic medium with that of the constituting single crystals and the function describing the orientation distribution of the crystals is derived. By expanding the orientation distribution function (ODF) into tensor spherical harmonics and using canonical components of the elasticity tensors, it is shown that the elastic tensor of the medium is completely determined by a finite number of expansion coefficients, namely those with harmonic degree l≤ 4. The number of expansion coefficients actually needed to determine the elastic constants of the medium depends on the symmetry of the single crystals. For hexagonal symmetry of the single crystals it is shown that only 8 real numbers are required to fix the 13 elastic constants which are for example needed to determine the azimuthal dependence of surface wave velocities. Thus, inversions of observations of seismic anisotropy are feasible which do not make any a priori assumptions on the orientation of the crystals. As a byproduct of the derivation, a formula is given which allows the easy calculation of the elastic constants of a medium composed of hexagonal crystals obeying an arbitrary ODF. An application of the theoretical results to the inversion of surface wave dispersion curves for an anisotropic 1D-mantle model is presented. For the S-wave velocities the results are similar to those of previous inversions but the new approach also yields P-wave velocities consistent with the assumption of oriented olivine. Moreover it provides a hint of the orientation distribution of the crystals.     

用应变地震观测求解震源矩张量的基本原理

震源机制解,即对地震矩张量的推断,对于地震研究具有至关重要的意义.应变地震观测是张量观测,与摆式地震仪的位移矢量观测不同,可以为地震研究提供新的数据源.本文讨论用应变地震观测求解震源矩张量的基本原理.在距离震源足够远的地方,地震波可以看成平面波,其性质决定于震源矩张量.假设平面地震波的应变张量可以由震源矩张量通过坐标变换计算得到,就可以通过观测应变地震波求解震源机制.这个假设至少对于双力偶震源机制是成立的.由此可以证明,在理想的无限介质中,只要有两个以上不同地点的应变地震波观测,就可以解出震源矩张量.这为解决震源机制问题提供了新的方法.目前的地震矩张量求解方法需要两方面的条件:或者需要很多观测点(例如体波反演),或者需要长周期地震波资料(例如面波反演).这些方法只适用于分析比较大的地震.对于小震,因为通常其震中周围不会有足够多的摆式地震仪观测点观测到其地震波,而地震波周期又短,难以利用传统方法给出可靠的震源机制解,所以只需少数观测点就能求解震源矩张量的新方法特别有意义.用应变地震观测求解震源机制,可以给出更为精确的结果.     

On Choosing Effective Elasticity Tensors Using a Monte-Carlo Method

A generally anisotropic elasticity tensor can be related to its closest counterparts in various symmetry classes. We refer to these counterparts as effective tensors in these classes. In finding effective tensors, we do not assume a priori orientations of their symmetry planes and axes. Knowledge of orientations of Hookean solids allows us to infer properties of materials represented by these solids. Obtaining orientations and parameter values of effective tensors is a highly nonlinear process involving finding absolute minima for orthogonal projections under all three-dimensional rotations. Given the standard deviations of the components of a generally anisotropic tensor, we examine the influence of measurement errors on the properties of effective tensors. We use a global optimization method to generate thousands of realizations of a generally anisotropic tensor, subject to errors. Using this optimization, we perform a Monte Carlo analysis of distances between that tensor and its counterparts in different symmetry classes, as well as of their orientations and elasticity parameters.     

Moment Tensor Inversion of Regional Phases: Application to a Mine Collapse

—?The aim of our study consists of analyzing potentially non-double-couple seismic events recorded at regional distances. In order to define the nature of the seismic source, a moment tensor inversion is carried out as this method is general enough not to initially constrain the source mechanism. In this paper we present an application to a seismic event induced by a mine collapse which occurred near the town of Halle in Germany. Because of its induced nature, many parameters such as the location and geometry of this seismic source are known. This information allows us to test the influence of inadequate propagation modeling on the moment tensor obtained from the inversion. Green's functions have been computed with the reflectivity method in a flat layered medium, using the European model EurID (Du et? al., 1998; Dufumier et al., 1997). From the inversion of P-wave seismograms recorded by the German Regional Seismic Network will, we obtained a source time function which can be decomposed into two subevents. The first one has a large isotropic part and a deviatoric mechanism with near vertical nodal planes. No volume change is observed for the second subevent, but a deviatoric component opposite of the first one. The addition of S-waves does not change the results of the inversion which are stable. Surface waves were not used because of their poor dispersion curves. Based on the moment tensor obtained from these inversions, the physical process at the source is compatible with a large cavity collapse.